Two Logarithmically Improved Regularity Criteria for the 3D Nematic Liquid Crystal Flows

Abstract

In this paper, we study the regularity criterion for the local smooth solution of the 3D nematic liquid crystal flows. More precisely, it is proved the smooth solution can be extended beyond provided that or .

1. Introduction

Liquid crystals is a state of the matter which has both properties of the liquid and the solid crystal. And as a kind of liquid crystals, the nematic liquid crystal can flow like fluids and has very nice properties. Ericksen et al. during 1960s (see [1, 2]) established the hydrodynamic theory for describing the nematic liquid crystal flows. Owing to the complexity of original Ericksen-Leslie equations and for further research, Lin [3] simplify the original Ericksen-Leslie equations, which still retains most of the essential features of original equations. In this paper, we investigate the following simplified version for nematic liquid crystal flows in 3-dimensionshere denotes the velocity field, (the unit sphere in ) the macroscopic average of molecular orientation field and represents the scalar pressure, is the incompressible condition. And , , are positive constants, which shall be assumed to be all equal to 1 in consideration of their concrete values playing no role in our arguments. The notation represents the matrix whose the component is given by

It is well-known that the system (1) has a unique local smooth solution (see [4]). More precisely, if initial data with and for , then

However, the global existence of solutions is an difficult problem. Hence much efforts have been paid to study the regularity criteria to extend local solutions. For the regularity criteria readers may refer to [5–13] and references therein.

On one hand, the above system (1) reduces to the incompressible Navier-Stokes equations when the orientation field equals a constant. It is well-known that Navier-Stokes equations has an unique smooth solution (see [14]) provided that the solution satisfieswhere . Later, Kozono and Taniuchi [15], Kozono et al. [16] generalized the criterion (4) torespectively, where is the space of Bounded Mean Oscillation and represents the homogeneous Besov space. And based on (5), Fan et al. [17], Guo and Gala [18] respectively improve (5) by the following conditions

It is obvious that the logarithmic improvement is here, in time only, and that can be seen as a natural Gronwall type extension of the Prodi-Serrin conditions. On the other hand, when the velocity field , the system (1) becomes to the heat flow of harmonic maps onto a sphere. And Wang [19] established a blow up criterion, which implies the unique smooth solution is global if

Inspired by the conditions (4) and (8), Huang and Wang [4] established a BKM type blow-up criterion for the system (1). That is, if is the maximal time, , then

Naturally, similar to (6) and (7), Liu and Zhao [20] extend (9) to the Logarithmically improved regularity criterion. Namely, the local smooth solution can continuously past any time if the following holdsor

In view of it is difficult to reduce the condition on , we are mainly concerned with reducing the condition on . Inspired by the references above, we will use the components of to replace the condition (11). Our main results are stated as follows:

Theorem 1. Assume is a local smooth solution to the system (1) on the time intervalfor some. And let initial datumwith,. Ifsatisfiesthen can be extended beyond smoothly, where . That is to say, if the solution blows up at , then

Remark 1. (1)In view of the fact that the norms and are approximate, it is obvious that the condition(10) is weaker than the condition (8) and (9) in some sense. And it can be seen that if the condition (10) reduces to then the conclusion of Theorem (1) still remains valid, which is also an improved result compared to the regularity criterion (9).(2)Noting that the norm is equivalent to , combining (10)–(12), the condition (12) can be replaced by the following condition:

Remark 2. It is well-known that . Thus the conclusion of Theorem 1still remains true if the condition (12) is substituted by

Theorem 2. Assume is a local smooth solution to the system (1) on the time intervalfor some. And let initial datumwith,. Ifsatisfieshere , then can be extended beyond smoothly. That is to say, if the solution blows up at , then

Remark 3. Owing to containing the case as , the condition (17) is an improvement in some sense compared to the condition (12). However,when, hence the condition (12) is better than the condition (17) in the end point.

2. Preliminaries

In this section, we collect some useful analytic tools which play an important part in our proof.

Lemma 1 (Page 82 in [21]). Let and be a positive real number. Then there exists a constant C such that

In particular, when , and , we have and

Lemma 2. (Product and Commutator estimate[22, 23]). Let,, andwith,and,. Then,where .

Lemma 3 (see [24], Lemma 2). Let,for all. Then there exists a positive constant C such thatwhere denotes the standard Sobolev space and

3. Proof of Main Results

Proof of Theorem 1. In this section, we shall first show the proof of Theorem 1. Since the existence of local smooth solutions is obvious owing to the initial value condition for in Theorem 1, we only need to show the priori estimate for the local smooth solution. And by the condition (12), we will give the following priori estimatewhich is enough to guarantee the smooth solution (, d) pasts time smoothly.
Firstly, we will show the estimate of and together because the terms and can be cancelled when integrating. Applying to the equation and integrating over yieldswhere the following equalities have been usedThen, multiplying equation by and integrating over one hasBy adding the above equalities and using the facts , we haveIntegrating above inequality (29) in time yieldsBesides, it is sufficient to show the boundness for . Multiplying equation by for and integrating both sides on , one haswhere we use the equalityHence the (31) impliesApplying the Gronwall inequality and letting , it can be deduced from above thatNow we shall show the estimate of and . Similarly going on the above process, multiplying the equation by and integrating over , then taking on the equation , multiplying with and integrating over , and combining that two equations, one obtains thatIn the following, we will estimate the terms . For , making use of the incompressibility condition and integration by parts several times, one can be concluded thatNoting that , and using the Lemma 3, can be estimated as followswhere we have used the following inequalityIn view of and containing terms that could be cancelled, adding and together and by the incompressibility condition , it follows thatHence, it can be deduced from H lder inequality, Young inequality and the inequality (20) thatwhere the fact that the norms and are equivalent has been used. For , by the product estimate (21) and inequality (20), we obtainhere the following Gagliardo-Nirenberg inequality has been usedCombining (35), (37), (39), and (41) one hasNoting (12), one may conclude that for any small constant , there exists such thatFor any , we setUsing (44) and (45), and applying Gronwall inequality to (43) in the interval giveswhere the letter means a constant depending on , depends on the maximum value of time , and is a generic constant which may be different from line to line.
At last, the boundedness of the norm and are needed so as to guarantee the validness of inequality (25) and (46). Employing and to the equations and respectively, and taking the inner product with , we see thatFor , applying , H lder inequality, the commutator estimate (22) and Young inequality, we havehere we have used the following Gagliardo-Nirenberg inequality:For , by the above inequalities used for and product estimate (21), we havehere we have used the following Gagliardo-Nirenberg inequalities:For , similar as (48) and by the above Gagliardo-Nirenberg inequalities, one may concludeFor , by the product estimate (21) and the fact , we infer thatInserting the above estimates (48), (50), (52), (53) to (47), and combining (46) yieldsIntegrating the above inequality with respect to time from to , , it follows thathere we choose . The above inequality and equality (45) imply thatTherefore, employing Gronwall’s inequality leads towhich indicates the truth of equality (25). Thus the Proof of Theorem 1 is completed.

Proof of Theorem 2. For the proof of Theorem 2, we only need reestimate again. By the Lemma 1, we haveGoing the same process to (39)–(57), the desired result will be obtained. Thus the Proof of Theorem 2 is completed.

Data Availability

There is no underlying data.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The fourth author Qiang Li is supported by the Xinyang College Research Projects, the grant numbers is 2022-XJLYB-004.